MHB How Do Vieta's Formulas Apply to Cubic Polynomial Roots?

Thread starter Mathsonfire
Start date Jul 12, 2018
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Cubic Polynomial

AI Thread Summary

Vieta's formulas provide relationships between the coefficients of a cubic polynomial and its roots, allowing for the calculation of sums and products of the roots. The expressions for the sums of pairs of roots, such as $(\alpha+\beta)+(\alpha+\gamma)+(\beta+\gamma)$, can be derived using these formulas. Additionally, the product of sums of pairs of roots, represented by $(\alpha+\beta)(\alpha+\gamma)(\beta+\gamma)$, can also be evaluated through Vieta's relationships. Another expression, $(\alpha+\beta)(\alpha+\gamma)+(\beta+\gamma)(\alpha + \beta)+(\alpha+\gamma)(\beta+\gamma)$, showcases further applications of Vieta's formulas in understanding cubic polynomials. Overall, these discussions highlight the utility of Vieta's formulas in analyzing the relationships between roots of cubic polynomials.

Jul 12, 2018

Mathsonfire

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Jul 13, 2018

MountEvariste

You can read $\alpha + \beta + \gamma $, $\alpha\beta +\alpha \gamma + \beta \gamma$ and $\alpha \beta \gamma$ off from the polynomial.

Now what's $(\alpha+\beta)+(\alpha+\gamma)+(\beta+\gamma)$?

And what's $(\alpha+\beta)(\alpha+\gamma)(\beta+\gamma)$?

And finally $(\alpha+\beta)(\alpha+\gamma)+(\beta+\gamma)(\alpha + \beta)+(\alpha+\gamma)(\beta+\gamma)$?Those questions all seem to be an exercise in Vietas formulas.

Last edited: Jul 13, 2018

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